Kertas 2 SPM Matematik Tambahan 2004-2010

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Disediakan oleh Mohamad Esmandi Bin Hapni

SECTION A

FUNCTIONS

1 Given that 23: xxf and 15

: x

xg , find

a) f -1

(x) [1 mark]P2,2006

b) f -1

g(x) [2 marks] P2,2006

c) h(x) such that hg(x) = 2x + 6 [3 marks] P2,2006

QUADRATIC FUNCTIONS

1 Diagram shows the curve of quadratic function f(x) = -x2 + kx – 5. The curve has a maximum point at

B(2, p) and intersects that f(x)-axis at point A.

a) State the coordinates of A [1 mark] P2,08

b) By using the method of completing the square, find

the value of k and of p. [4 marks] P2,08

2 The quadratic equation 0652 xx has roots h and k, where h > k.

a) Find

i) the value of h and of k.

ii) the range of x if 0652 xx [5 marks]P2,2009

b) Using the value of h and k from (a i), form the quadratic equation which has roots h + 2 and 3k – 2

[2 marks]P2,2009

SIMULTANEOUS EQUATIONS

1 Solve the simultaneous equations p – m = 2 and p2 + 2m = 8

Give your answers correct to three decimal places. [5 marks]P2,2004

2 Solve the simultaneous equations 12

1 yx and y

2 – 10 = 2x [5 marks]P2,2005

3 Solve the simultaneous equations 2x + y = 1 and 2x2 + y

2 + xy = 5 [5 marks]P2,2006

4 Solve the simultaneous equations ;

2x – y – 3 = 0, 2x2 – 10x + y + 9 = 0 [5 marks]P2,2007

5 Solve the following simultaneous equations :

x – 3y + 4 = 0

x2 + xy – 40 = 0 [5 marks]P2,2008

O

f(x)

x

B(2, p)

A

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6 Solve the simultaneous equation k – 3p = -1 and p + pk – 2k = 0. Give your answer correct to three

decimal places. [5 marks]P2,2009

7 Solve the simultaneous equation x – 2y = 7 and xy – x = 9y. Give your answer correct to two decimal

places. [5 marks]P2,2010

COORDINATE GEOMETRY

1 Diagram shows a straight line CD which meets a straight line AB at the point D. The point C lies on

the y-axis.

a) Write down the equation of AB in the form of intercepts

[1 mark]P2,2004

b) Given that 2AD = DB, find the coordinates of D

[2 marks]P2,2004

c) Given that CD is perpendicular to AB, find the y-intercept of CD

[3 marks]P2,2004

2 Solution by scale drawing will not be accepted.

In diagram, the straight line AB has an equation y + 2x + 8 = 0. AB intersects the x-axis at point A and

intersects the y-axis at point B.

Point P lies on AB such that AP : PB = 1 : 3, Find

a) the coordinates of P [3 marks]P2,2007

b) the equation of the straight line that passes through P and

perpendicular to AB [3 marks]P2,2007

3 Solution by scale drawing will not be accepted.

Diagram shows the straight line AC which intersects the y-axis at point B.

B(9, 0)

A(0, -6)

D

O x

y

C

A

P

B

O x

y

y + 2x + 8 = 0

O

y C

B

A(-3 , -7)

x

The equation of AC is 3y = 2x – 15

Find

(a) the equation of the straight line which passes through

point A and perpendicular to AC.

[4 marks]P2,2010

(b) (i) the coordinates of B,

(ii) the coordinates of C, given AB : BC = 2 : 7

[3 marks’P2,2010

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STATISTICS

1 A set of data consists of 10 numbers. The sum of the numbers is 150 and the sum of the squares of the

number is 2 472.

a) Find the mean and variance of the 10 numbers [3 marks]P2,2004

b) Another number is added to the set of data and the mean is increased by 1. Find

i. the value of this number,

ii. the standard deviation of 11 numbers. [4 marks]P2,2004

2 Diagram shows a histogram which represents the distribution of the marks obtained by 40 pupils in a

test.

a) Without using ogive, calculate the median mark.

[3 marks]P2,2005

b) Calculate the standard deviation of the distribution.

[4 marks]P2,2005

3 Table shows the frequency distribution of the scores of a group of pupils in a game.

a) It is given that the median score of the distribution is 42.

Calculate the value of k.

[3 marks]P2,2006

b) Use the graph paper to answer this question.

Using a scale of 2 cm to 10 scores on the horizontal axis

and 2 cm to 2 pupils on the vertical axis, draw a histogram

to represent the frequency distribution of the scores. Find

the mode score.

[4 marks]P2,2006

c) What is the mode score if the score of each pupil is

increased by 5?

[1 mark]P2,2006

4 Table shows the cumulative frequency distribution for the scores of 32 students in a competition.

Score < 10 < 20 < 30 < 40 <50

Number of students 4 10 20 28 32

a) Based on the table above, copy and complete table below

Score 0 - 9 10 - 19 20 - 29 30 -39 40 – 49

Number of students

[1 mark]P2,2007

b) Without drawing an ogive, find the interquartile range of the distribution.

[5 marks]P2,2007

0

2

4

6

8

10

12

14

Number of Pupils

0.5 10.5 20.5 30.5 40.5 50.5 Marks

Score Number of Pupils

10-19 1

20-29 2

30-39 8

40-49 12

50-59 k

60-69 1

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5 Table shows the marks obtained by 40 candidates in a test.

Given that the median mark is 35.5, find the value of x and of y.

Hence, state the modal class.

[6 marks]P2,2008

6 Table shows the frequency distribution of the marks of a group of students.

Marks Number of students

1 – 10 5

11 – 20 8

21 – 30 20

31 – 40 10

41 - 50 7

PROGRESSIONS

1 triangle has a base of x cm and a height of y cm. The measurement of the base and height of each

subsequent triangle are half of the measurements of its previous one.

a) Show that the areas of the triangles form a geometric progression

and state the common ratio

[3 marks]P2,2004

b) Given that x = 80 cm and y = 40 cm,

i) determine which triangle has an area of 4

16 cm

2,

ii) find the sum to infinity of the areas, in cm2, of the triangles.

[5 marks]P2,2004

Marks Number of

candidates

10 – 19 4

20 – 29 x

30 – 39 y

40 – 49 10

50 - 59 8

(a) Use graph paper to answer this part of the question.

Using scale of 2 cm 10 marks on the horizontal axis

and 2 cm to 2 students on the vertical axis, draw a

histogram to represent the frequency distribution of

the marks in the table.

Hence find the mode mark.

[4 marks]P2,2010

(b) Calculate the standard deviation of the marks.

[4 marks]P2,2010

y cm

x cm

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2 Diagram shows part of an arrangement of bricks of equal size.

The number of bricks in the lowest row is 100. For each of the other rows, the number of bricks is 2

less than in the row below. The height of each brick is 6 cm. Ali builds a wall by arranging bricks in

this way. The number of bricks in the highest row is 4. Calculate

a) the height in cm of the wall. [3 marks]P2,2005

b) the total price of the bricks used if the price of one brick is 40 sen. [3 marks]P2,2005

3 Two companies, Delta and Omega, start to sell cars at the same times

a. Delta sells k cars in the first month and its sales increases constantly by m cars every

subsequent month. It sells 240 cars in the 8th month and the total sales for the first 10 months

are 1900 cars. Find the value of k and of m.

[5 marks]P2,2006

b. Omega sells 80 cars in the first month and its sales increases constantly by 22 cars every

subsequence month. If both companies sell the same number of car in the nth month, find the

value of n.

[2 marks]P2,2006

4 Diagram shows the side elevation of part of stairs built of cement blocks.

The thickness of each block is 15 cm. The length of the first block is 985 cm. The length of each

subsequence block is 30 cm less then preceding block as shown in diagram.

a) If the height of the stairs to built is 3 m, calculate

i) the length of the top most block

ii) the total length of the blocks.

[5 marks]P2,2007

b) Calculate the maximum height of the stairs.

[3 marks]P2,2007

6 cm

985 cm

955 cm

925 cm

15 cm

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5 Muthu starts working for a company on 1 January 2002 with an initial annual salary of RM18 000.

Every January, the company increased his salary 5% of the previous year’s salary. Calculate

a. his annual salary to nearest RM for the year 2007

[3 marks]P2,2008

b. the minimum value of n such that his annual salary in the nth year will exceed RM36 000.

[2 marks]P2,2008

c. the total salary, to the nearest RM, paid to him by the company for the years 2002 to 2007.

[2 marks]P2,2008

6 Amir drops a ball from a height of H cm above the floor. After the first bounce, the ball reaches a

height of H1 cm, where H1 = 0.8 H. After the second bounce, the ball reaches a height of H2 cm, where

H2 = 0.8 H1. The ball continues bouncing in this way until it stops.

Given that H = 200, find

a. the number of bounces when the maximum height of the ball from the floor is less than 50 cm

for the first time,

[4 marks]P2,2009

b. the total distance, in cm, traveled by the ball until it stops.

[2 marks]P2,2009

7 Diagram shows the arrangement of cylinders having the same radius, 3 cm. The height of the first

cylinder is 4 cm and the height of each subsequent cylinder increases by 2 cm.

[Volume of cylinder = r2 h]

(a) Calculate the volume, in cm3, of the 17

th cylinder, in term of .

[3 marks]P2,2010

(b) Given the total volume of the first n cylinders is 1620 cm3, find the value of n.

(c) [3 marks]P2,2010

4 cm 6 cm 8 cm

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VECTORS

1 Diagram shows triangle OAB. The straight line AP intersects the straight line OQ at R. It is given that

OP = OB3

1, AQ = AB

4

1,

~6 xOP and

~

2 yOA

a) Express in term of ~x and/or

~

y

i) AP

ii) OQ [4 marks]P2,2004

b) (i) Given that APhAR , state AR in terms of h, ~x

and ~

y

(ii) Given that OQkRQ , state RQ in terms of k,

~x and

~

y

[2 marks]P2,2004

c) Using AR and RQ from (b) find the value of h and

of k.

[4 marks]P2,2004

2 In the diagram, ABCD is a quadrilateral. AED and EFC are straight lines

It is given that ~

20 xAB , ~

8 yAE , ~~

2425 yxDC ,

ADAE4

1 and ECEF

5

3

a) Express in terms of ~x and/or

~

y

i) BD

ii) EC [3 marks]P2,2005

b) Show that the points B, F and D are collinear.

[3 marks]P2,2005

c) If ~x = 2 and

~

y = 3, find BD

[2 marks]P2,2005

R

Q

A

P O B

A B

C

D

E F

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3 Diagram shows a trapezium ABCD.

It is given that ~

2 yAB , ~

6 xAD , ADAE3

2 and

ADBC6

5

a) Express AC , in terms of ~x and

~

y . [2 marks]P2,2006

b) Point F lies inside the trapezium ABCD such that

ABmEF 2 , and m is a constant.

i) Express AF , in terms of m, ~x and

~

y .

ii) Hence, if the point A, F and C are collinear, find the value

of m.

[5 marks]P2,2006

4 In diagram ABCD is a quadrilateral. The diagonal BD and AC intersects at point R. Point P lies on AD.

It is given that ADAP3

1 , BDBR

3

1 ,

~xAB and

~

yAP

a) Express I terms of ~x and/or

~

y :

i) DB

ii) AR [3 marks]P2,2008

b) Given that ~~yxkDC and AChAR , where h and

k are constants, find the value of h and if k.

[4 marks]P2,2008

5 Diagram shows triangle ABC. The straight line AQ intersects the straight line BR at P.

It is given that AR = 3RC, BQ = 3

2BC, xAB 3 and yAC 4


~

y :

i) BC

ii) AQ [3 marks]P2,2009

b) It is given that AQhAP and RBkARAP , where h and

k are constants. Find the value of h and of k

[5 marks]P2,2009

A

B C

D

F

E

A

P

B

R

C

D

A B

Q

P

R

C

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TRIGONOMETRIC FUNCTIONS

1 (a) Sketch the graph of y = cos2x for 0o x 180

o.

[3 marks]P2,2004

(b) Hence, by drawing a suitable straight line on the same axes, find the number of solutions

satisfying the equation 180

2sin2 2 xx for 0

o x 180

o.

[3 marks]P2,2004

2 (a) Prove that xxxxec 2coscotsin2cos 222

[2 marks]P2,2005

(b)(i) Sketch the graph of y = cos2x for 0 x 2

(ii) Hence, using the same axes, draw a suitable straight line to find the number of solutions to the

equation 1cotsin2cos3 222

xxxxec for 0 x 2. State the number of solutions.

[6 marks]P2,2005

3 (a) Sketch the graph of xy cos2 for 0 x 2

[4 marks]P2,2006

(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the

equation 0cos2 xx

for 0 x 2. State the number of solutions.

[3 marks]P2,2006

4 (a) Sketch the graph of xy 2cos3 for 0 x 2

[4 marks]P2,2007

(b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the

equation 2

2cos32x

x for 0 x 2. State the number of solutions.

[3 marks]P2,2007

5 (a) Prove that xx

x2tan

sec2

tan22

. [2 marks]P2,2008

(b)(i) Sketch the graph of xy 2tan for 0 x

(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for

the equation 0sec2

tan232

x

xx

, for 0 x

[6 marks]P2,2008

6 (a) Sketch the graph of xy 2cos2

3 for

2

30 x . [3 marks]P2,2009


equation 2

32cos

3

4 xx

for

2

30 x . State the number of solutions.

[3 marks]P2,2009

7 (a) Sketch the graph of y = 1 + 3cos x for 0 ≤ x ≤ 2. [4 marks]P2,2010


equation 6 cos x = 4 - 3x for 0 ≤ x ≤ 2.

[3 marks]P2,2010

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DIFFERENTIATION AND INTEGRATION

1 The gradient function of the curve which passes through A(1, -12) is xx 63 2 , Find

a) the equation of the curve [3 marks]P2,2004

b) the coordinates of the turning points of the curve and determine whether each of the turning points

is a maximum or a minimum. [5 marks]P2,2004

2 The gradient function of a curve is kxhx 2 , where h and k are constants. The curve has a turning

point at (3, -4). The gradient of the tangent to the curve at the point x = -1 is 8. Find

a) the value of h and of k [5 marks]P2,2009

b) the equation of the curve [3 marks]P2,2009

3 Diagram shows the shaded region bounded by the curve y = g(x), the x-axis and the straight line x = 3.

O 1

y

x

x = 3

y = g(x)

It is given that the area of the shaded region is 10

unit2.

a) Find

i) 3

1)( dxxg

ii) 3

1)](2[ dxxgx

[4 marks]P2,2010

b) Given g(x) = 2x – 8, find g(x) in terms of x.

[3 marks]P2,2010

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SECTION B

COORDINATE GEOMETRY

1 Solution to this question by scale drawing will not be accepted.

a. Find

i. the equation of straight line AB

ii. the coordinates of B

[5 marks]P2,2005

b. The straight line AB is extended to a point D such that

AB : BD = 2 : 3, Find the coordinates D.

[2 marks]P2,2005

c. A point P moves such that its distance from point A is

always 5 units. Find the equation of the locus of P.

[3 marks]P2, 2005

2 Solution to this question by scale drawing will not be accepted.

a. Calculate the area, in unit2, of triangle AOB

[2 marks]P2, 2006

b. Given that AC : CB = 3 : 2, find the coordinates of C.

[2 marks]P2,2006

c. A point P moves such that its distance from point A is

always twice its distance from point B.

i. Find the equation of the locus of P.

ii. Hence, determine whether or not this locus

intercepts the y-axis.

[6 marks}P2,2006

3 Solution by scale drawing is not accepted.

Diagram shows a triangle OPQ. Point S lies on the line PQ.

a. A point W moves such that its distance from point S is always

2

12 units. Find the equation of the locus of W.

[3 marks]P2,2008

b. It is given that point P and point Q lie on the locus of W.

Calculate

1. The value of k

2. The coordinates of Q

[5 marks]P2,2008

c) Hence, find the area, in unit2, of triangle OPQ

[2 marks]P2,2008

B

A(-4, 9)

C

x

y

O 2y + x + 6 = 0

B(6, -2)

A(-3, 4)

C

O x

y

y

x

P(2

3, k)

S(3, 1)

Q

O

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4 Solution by scale drawing is not accepted.

Diagram shows a trapezium OABC. The line OA is perpendicular to the line AB, which intersects

the y-axis at the point Q. It is given that the equation of OA is xy2

3 and the equation of AB is

266 kxy

a) Find

i) the value of k

ii) the coordinates of A

[4 marks]P2,2009

b) Given AQ : QB = 1 : 2, find

i) the coordinates of B

ii) the equation of the straight line BC

[4 marks]P2,2009

c) A point P(x, y) moves such that 2PA = PB

Find the equation of the locus of P.

[2 marks]P2,2009

CIRCULAR MEASURE

1 Diagram shows a circle PQRT, centre O and radius 5 cm. JQK is a tangent to the circle at Q. The

straight lines, JO and KO, intersect the circle at P and R respectively. OPQR is a rhombus. JLK is an

arc of a circle O.

Calculate

a) the angle , in terms of

[2 marks]P2,2004

b) the length, in cm, of the arc JLK

[4 marks]P2,2004

c) the area, in cm2, of the shaded region.

[4 marks]P2,2004

2 Diagram shows a sector POQ of a circle, centre O. The point A lies on OP, the point B lies on OQ and

AB is perpendicular to OQ. The length of OA = 8 cm and 6

POQ radian.

It is given that OA : OP = 4 : 7

(Use = 3.142)

Calculate

a) the length in cm of AP [1 mark ]P2,2005

b) the perimeter in cm of the shaded region [5 marks]P2,2005

c) the area in cm2 of the shaded region [4 marks]P2,2005

O

A

Q

B

C

x

y

xy

2

3

266 kxy

rad

O

R P

Q J K

L

T

P

Q B O

A

8 cm

rad6

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3 Diagram shows the plan of a garden. PCQ is a semicircle with centre O and has a radius of 8 m. RAQ

is a sector of the circle with centre A and has a radius of 14 m.

Sector COQ is a lawn. The shaded region is a flower bed

and has to be fenced. It is given that AC = 8 m and

COQ = 1.956 radians.

Calculate

a) the area in m2 of the lawn

[2 marks]P2,2006

b) the length in m, of the fence required for fencing the

flower bed.

[4 marks]P2,2006

c) the area in m2, of the flower bed.

[4 marks]P2,2006

4 Diagram shows a circle, centre O and radius 10 cm inscribed on a sector APB of a circle, centre P. The

straight lines, AP and BP, are tangents to the circle at point Q and point R, respectively.

(Use = 3.142)

Calculate

a) the length in cm of the arc AB

[5 marks]P2,2007

b) the area in cm2 of the shaded region.

[5 marks]P2,2007

5 Diagram shows two circles. The larger circle has centre X and radius 12 cm. The smaller circle has

centre Y and radius 8 cm. The circles touch at point R. The straight line PQ is a common tangent to the

circles at point P and point Q.

(Use = 3.142)

Given that PXR = radians,

a) show that = 1.37 (to two decimals places)

[2 marks]P2,2008

b) calculate the length, in cm, of the minor arc QR

[3 marks]P2,2008

c) calculate the area, in cm2, of the shaded region

[5 marks]P2,2008

O Q A P

C

R

O

A B

Q R

10 cm

60o

P

Y

X

P

Q

R

12 cm

8 cm

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6 In diagram, POQ is a sector of a circle with centre O and radius 26 cm. SRT is a quadrant of a circle

with centre R and radius 5 cm.

It is given that S is the midpoint of OP. Use = 3.142 and give the answers correct to two decimal

places.

Calculate

a) POQ, in radian

[2 marks]P2,2009

b) the perimeter in cm of the shaded region.

[4 marks]P2,2009

c) the area in cm2 of the shaded region.

[4 marks]P2,2009

7 Diagram shows a circle with centre O. D is the midpoint of AC and DOC = 60o.

DIFFERENTIATION

1 The curve y = x3 – 6x

2 + 9x + 1 passes through the point A(2 , 3) and has two turning points, P(3 , 1)

and Q.

Find

(a) the gradient of the curve at A. [3 marks]P2,2010

(b) the equation of the normal to the curve at A. [3 marks]P2,2010

(c) the coordinates of Q and determine whether Q is the maximum or the minimum point.

[4 marks]P2,2010

O R T Q

P

S

A C

B

D

O

60o

It is given that the area of the major sector OAC is

15

7209 cm

2.

[Use = 3.142]

Find

(a) the value of , in radians, [1 mark]P2,2010

(b) radius, in cm, of the circle, [2 marks]P2,2010

(c) perimeter, in cm, of the shaded sector

[4 marks]P2,2010

(d) the area in cm2, of segment ABC

[3 marks]P2,2010

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4 Diagram shows part of the curve 212

3

xy which passes through A(1, 3)

a. Find the equation of the tangent to the curve at point

A.

[4 marks]P2,2004

b. A region is bounded by the curve, the x-axis and the

straight lines x = 2 and x = 3.

i. Find the area of the region

ii. The region is revolved through 360o about the x-

axis. Find the volume generated, in term of .

[6 marks]P2,2004

5 The diagram shows, the straight line PQ is normal to the curve 12

2

x

y at A(2, 3). The straight line

AR is parallel to the y-axis.

Find

c. the value of k

[3 marks]P2,2006

d. the area of the shaded region

[4 marks]P2,2006

e. the volume generated, in terms of , when the region

bounded by the curve, the y-axis and the straight line

y = 3 is revolved through 360o about the y-axis.

[3 marks]P2,2005

6 The curve with gradient function 2

22

xx has a turning point at (k, 8).

f. Find the value of k. [3 marks]P2,2007

g. Determine whether the turning point is a maximum or a minimum point. [2 marks]P2,2007

h. Find the equation of the curve. [3 marks]P2,

2007

A(1, 3)

212

3

xy

x

y

O

R O Q(k, 0)

x

A(2, 3)

P

y

12

2

x

y

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7 Diagram shows part of the curve 31 xky , where k is a constant.

The curve intersects the straight line x = 3 at point

A.

At point A, dx

dy= 24

a) Find the value of k

[3 marks]P2,2007

b) Hence, calculate

i) the area of shaded region P.

ii) the volume generated in term of when

the region R which is bounded by the

curve, the x-axis and the y-axis, is

revolved through 360o about x-axis.

[7 marks]P2,2007

8 Diagram shows the curve 52 xy and the tangent to the curve at the point A(1, 6).

Calculate

i. the equation of the tangent at A.

[3 marks]P2,2008

j. the area of the coloured region,

[4 marks]P2,2008

k. the volume of revolution, in term of , when the

region bounded by the curve and the straight line y

= 7 is rotated through 360o about the y-axis.

[3 marks]P2,2008

LINEAR LAW

1 Table shows the values of two variables, x and y, obtained from an experiment. Variables x and y are

related by the equation y = pkx, where p and k are constants.

x 2 4 6 8 10 12

y 3.16 5.50 9.12 16.22 28.84 46.77

a) Plot y10log against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit on the

y10log -axis.

[4 marks]P2,2004

b) Use your graph from (a) to find the value of

i) p

ii) k

[6 marks]P2,2004

P

A

R O

y

x

x = 3

31 xky

O x

y

52 xy

A(1, 6)

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2 Table shows the values of two variables, x and y, obtained from an experiment. The variables x and y

related by the equation px

rpxy , where p and r are constants.

x 1.0 2.0 3.0 4.0 5.0 5.5

y 5.5 4.7 5.0 6.5 7.7 8.4

a) Plot xy against x2, by using scale of 2 cm to 5 units on both axes. Hence, draw the line of best fit.

[5 marks]P2,2005

b) Use the graph from (a) to find the value of

i) p

ii) r

[5 marks]P2,2005

3 related by the equation y = pkx+1

, where p and k are constants.

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

a) Plot y10log against (x + 1) using a scale 2 cm to 1 unit on (x + 1)-axis and 2 cm to 0.2 unit on the

y10log -axis. Hence, draw the line of the best fit.

[5 marks]P2,2006

b) Use your graph from (a) to find the value of

i) p,

ii) k

[5 marks]P2,2006


related by the equation xk

pkxy 22 , where p and k are constants.

x 2 3 4 5 6 7

y 8 13.2 20 27.5 36.6 45.5

a) Plot x

y against x, using a scale of 2 cm to 1 unit on both axes. Hence draw the line of best fit.

[4 marks]P2,2007

b) Use your graph in (a) to find the value of

i) p

ii) k

iii) y when x = 1.2

[6 marks]P2,2007

P a g e | 18



related by the equation y = hk2x

, where h and k are constants.

x 1.5 3.0 4.5 6.0 7.5 9.0

y 2.51 3.24 4.37 5.75 7.76 10.00

a) Based on the table, construct a table for the values of y10log .

[1 mark]P1,2008

b) Plot y10log against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 0.1 unit on the

y10log -axis. Hence, draw the line of best fit.

[4 marks]P2,2008

c) Use the graph in (b) to find the value of

i) x when y = 4.8

ii) h

iii) k

[5 marks]P2,2008

6 Table shows the values of two variables, x and y, obtained from an experiment. The variables x and y

are related by the equation 1x

p

y

k, where k and p are constants.

x 1.5 2.0 3.0 4.0 5.0 6.0

y 2.502 0.770 0.465 0.385 0.351 0.328

a) Based on the table, construct a table for the values of x

1 and

y

1.

[2 marks]P2,2009

b) Plot y

1 against

x

1, using a scale of 2 cm to 0.1 unit on

x

1-axis and 2 cm to 0.5 unit on

y

1-axis.

Hence, draw the line of best fit.

[3 marks]P2,2009

c) Use the graph in (b) to find the value of

i) k

ii) p

[5 marks]P2,2009

7 Use graph paper to answer this question.

Table shows the value of two variables, x and y, obtained from an experiment. Variables x and y are

related by equation k

hy

x

, where h and k are constants.

x 3 4 5 6 7 8

y 2.57 3.31 4.07 4.90 6.31 7.94

P a g e | 19


(a) Plot y10log against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 0.1 unit on the

y10log -axis.

Hence, draw the line of best fit.

[4 marks]P2,2010

(b) Use the graph in (a) to find the value of

(i) h,

(ii) k,

(iii) y when x = 2.7

[4 marks]P2,2010

VECTOR

1 Diagram shows triangle AOB. The point P lies on OA and the point Q lies on AB. The straight line BP

intersects the straight line OQ at the point S.

It is given that

OA : OP = 4 : 1, AB : AQ = 2 : 1, ~

8 xOA , ~

6 yOB


~

y

i) BP

ii) OQ

[3 marks]P2,2007 { [3 marks]P2,2007

b) Using OQhOS and BPkBS , where h and k are

constants. Find the value of h and of k

[5 marks]P2,2007

c) Given that 2~x units, 3

~

y units and AOB = 90o, find

AB

[2 marks]P2,2007

2 Diagram shows triangle OAB. The point C lies on OA and the point D lies on AB. The straight line OD

intersects the straight line BC at the point E.

O P A

Q

B

S

O C A

E

B

D

It is given that xOA , yOB , OAOC3

2 and

ADAB 2

(a) Express in terms of x and y

(i) BC

(ii) OD [4 marks]P2,2010

P a g e | 20


(b) It is given that ODhOE and BCkBE ,where h and k are constants.

Express OE

(i) in terms of h, x and y

(ii) in terms of k, x and y [3 marks]P2,2010

(c) Hence, find the value of h and of k. [3 marks]P2,2010

PROBABILITY DISTRIBUTIONS

1 (a) A club organises a practice session for trainees on scoring goals from penalty kicks. Each

trainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick

is p. After the session, it is found that the mean number of goals for a trainee is 4.8.

i) Find the value of p.

ii) If a trainee is chosen at random, find the probability that he scores at least one goal.

[5 marks]P2,2004

(b) A survey on body-mass is done on a group of students. The mass of a student has a normal

distribution with a mean of 50 kg and a standard deviation of 15 kg.

i) If a student is chosen at random, calculate the probability that his mass is less than 41

kg.

ii) Given that 12% of the students have a mass of more than m kg, find the value of m.

[5 marks]P2,2004

2 For this question, give your answer correct to three significant figures.

(a) The result of a study shows that 20% of the pupils in a city cycle to school. If 8 pupils from the

city is chosen at random, calculate the probability that

i) exactly 2 of them cycle to school.

ii) Less that 3 of them cycle to school.

[4 marks]P2,2005

(b) The mass of water-melons produced from an orchard follows a normal distribution with a mean

of 3.2 kg and a standard deviation of 0.5 kg. Find

i) the probability that a water-melons chosen randomly from the orchard has a mass of not

more than 4.0 kg.

ii) the value of m if 60% of the water-melons from the orchard have a mass of more than m

kg.

[6 marks]P2,2005

P a g e | 21


3 An orchard produces lemons. Only lemons with diameter, x greater than k cm are graded and

marketed.

Table shows the grades of the lemons based on their diameters.

Grade A B C

Diameter, x cm x > 7 7 x > 5 5 x > k

It is given that the diameter of the lemons has a normal distribution with mean of 5.8 cm and a

standard deviation of 1.5 cm.

a) If one lemon is picked at random, calculate the probability that it is of grade A.

[2 marks]P2,2006

b) In a basket of 500 lemons, estimate the number of grade B lemons.

[4 marks]P2,2006

c) If 85.7% of the lemons is marked, find the value of k.

[4 marks]P2,2006

4 (a) In a survey carried out in a school, it is found that 2 out of 5 students have handphones. If 8

students from that school are chosen at random, calculate the probability that

i) exactly 2 students have handphones,

ii) more than 2 students have handphones.

[5 marks]P2,2007

(b) A group of workers are given medical check up. The blood pressure of a worker has a normal

distribution with mean of 130 mmHg and a standard deviation of 16 mmHg. Blood pressure

that is more than 150 mmHg is classified as “high blood pressure”.

i) A worker is chosen at random from the group. Find the probability that the worker has

a blood pressure between 114 mmHg and 150 mmHg.

ii) It is found that 132 workers have “high blood pressure”. Find the total number of the

workers in the group.

[5 marks]P2,2007

5 The masses of mangoes from an orchard has a normal distribution with a mean of 300 g and a standard

deviation of 80 g.

a) Find the probability that a mango chosen randomly from this orchard has a mass of more than 168

g.

[3 marks]P2,2008

b) A random sample of 500 mangoes is chosen.

i) Calculate the number of mangoes from this sample that have a mass of more than 168 g.

ii) Given that 435 mangoes from this sample have a mass of more than m g, find the value of m.

[7 marks]P2,2008

6 A test paper consists of 40 questions. Each question is followed by four choices of answer, where only

one of these is correct.

a) Salma answers all the question by randomly choosing an answers for each question.

i) Estimate the number of questions she answered correctly.

ii) Find the standard deviation of the number of questions she answered correctly

[5 marks]P2,2009

P a g e | 22


b) Basri answered 30 questions correctly and randomly choosing an answer for each of the remaining

10 questions. Find the probability that he answers

i) 36 questions correctly,

ii) At least 32 questions correctly.

[5 marks]P2,2009

7 (a) A random variable, X, has a binomial distribution with 10 trials where the probability of success

in each trial is p. The mean number of success is 4.

Calculate

(i) The value of p,

(ii) P(x ≤ 2) [5 marks]P2,2010

(b) The diameter of limes from a farm has a normal distribution with mean of 3.2 cm and a standard

deviation of 1.5 cm.

Calculate

(i) the probability that a lime chosen at random from this farm has a diameter more than 3.9 cm.

(ii) the value of d if 33% of the limes have diameters less than d cm.

[5 marks]P2,2010

P a g e | 23


SECTION C

SOLUTION OF TRIANGLES

1 Solution by scale drawing will not be accepted

Diagram shows a quadrilateral ABCD such that ABC is acute.

a) Calculate

i) ABC

ii) ADC

iii) The area in cm2 of quadrilateral ABCD.

[8 marks]P2,2004

b) A triangle A’B’C’ has the same measurement as those given

for triangle ABC, that is A’C’ = 12.3 cm, C’B’ = 9.5 cm,

and B’A’C’ = 40.5o, but which is different in shape to

triangle ABC.

i) Sketch the triangle A’B’C’

ii) State the size of A’B’C’

[2 marks]P2,2004


Diagram shows triangle ABC

a) Calculate the length in cm of AC.

[2 marks]P2,2005

b) A quadrilateral ABCD is now formed so that AC is a diagonal, ACD =

40o and AD = 16 cm. Calculate the two possible values of ADC.

[2 marks]P2,2005

c) By using the acute ADC from (b), calculate

i) the length in cm of CD

ii) the area in cm2 of the quadrilateral ABCD.

[6 marks]P2,2005


Diagram shows a quadrilateral ABCD

The area of triangle BCD is 13 cm2 and BCD is acute.

Calculate

a) BCD [2 marks]P2,2006

b) the length in cm of BD [2 marks]P2,2006

c) ABD [3 marks]P2,2006

d) the area in cm2, quadrilateral ABCD. [3 marks]P2,2006

B

A

C

D

9.8 cm 12.3 cm

5.2 cm

9.5 cm 40.5

o

A

B

C

20 cm

15 cm

65o

A

B

C

D

40o

9 cm

6 cm

5 cm

P a g e | 24



Diagram shows quadrilateral ABCD

a) Calculate

i) the length in cm of AC

ii) ACB

[4 marks]P2,2007

b) Point A’ lies on AC such that A’B = AB

i) Sketch A’BC

ii) Calculate the area in cm2 of A’BC

[6 marks]P2,2007


In diagram below, ABC is a triangle. ADFB, AEC and BGC are straight lines. The straight line FG is

perpendicular to BC.

It is given that BD = 19 cm, DA = 16 cm, AE = 14

cm, DAE = 80o and FBG = 45

o.

a) Calculate

i) DE

ii) EC

[5 marks]P2,2008

b) The area of triangle DAE is twice the area of

triangle FBG. Calculate the length in cm of

BG.

[4 marks]P2,2008

c) Sketch triangle A’B’C’ which has different

shape from triangle ABC such that A’B’ = AB,

A’C’ = AC and A’B’C’ = ABC.

[1 mark ]P2,2008


Diagram shows a trapezium KLMN. KN is parallel to LM and LMN is obtuse.

Find

a) the length in cm of LN [2 marks]P2,2009

b) the length in cm of MN [3 marks]P2,2009

c) LMN [3 marks]P2,2009

d) the area in cm2 of triangle LMN [2 marks]P2,2009

A

B

C D 6 cm

16.4 cm

5.6 cm

105o

50o

45o

80o

A

B C

D E

F

G

80o

32o

K

N

M

L

5.6 cm

12.5 cm

P a g e | 25



Diagram shows triangle ABC and triangle CDE where BCE and ACD are straight lines.

INDEX NUMBER

1 Table shows the price indices and percentage of usage of four items P, Q, R, and S, which are the main

ingredients in the production of a type of biscuit.

Item Price index for the year 1995

based on the year 1993

Percentage of usage

(%)

P 135 40

Q x 30

R 105 10

S 130 20

a) Calculate

i) the price of S in the year 1993 if its price in the year 1995 is RM37.70,

ii) the price index of P in the year 1995 based on the year 1991 if its price index in the year

1993 based on the year 1991 is 120

[5 marks]P2,2004

b) The composite index number of the cost of biscuit production for the year 1995 based on the year

1993 is 128.

Calculate

i) the value of x

ii) the price of a box of biscuit in the year 1993 if the corresponding price in the year 1995 is

RM32

[5 marks]P2,2004

25o

50o

A

B C

D

E 6 cm

4 cm

2.5 cm

(a) Calculate the length, in cm, of

(i) BC

(ii) DE

[5 marks]P2,2010

(b) Point C lies on BE such that AC = AC

(i) Sketch triangle A C B.

(ii) Find A C B.

(iii)Calculate the area, in cm2, of triangle

AC B.

[5 marks]P2,2010

P a g e | 26


2 Table shows the price and the price indices for the four ingredients, P, Q, R, and S, used in making

biscuit of a particular kind. Diagram is a pie chart which represents the relative amount of the

ingredients P, Q, R, and S, used in making these biscuits.

Ingredients

Price per kg

(RM)

Price index for the year

2004 based on the year

2001 Year 2001 Year 2004

P 0.80 1.00 x

Q 2.00 y 140

R 0.40 0.60 150

S z 0.40 80

a) Find the value of x, y, and z [3 marks]P2,2005

b) (i) Calculate the composite index for the cost of making these biscuits in the year 2004 based on

the year 2001

(ii) Hence, calculate the corresponding cost of making these biscuits in the year 2001 if the cost in

the year 2004 was RM2985.

[5 marks]P2,2005

c) The cost of making these biscuits is expected to increase by 50% from the year 2004 to the year

2007. Find the expected composite index for the year 2007 based on the year 2001.

[2 marks]P2,2005

P

Q

R

S 100o

60o

120o

P a g e | 27


3 A particular kind of cake is made by using four ingredients P, Q, R, and S. Table shows the prices of

the gradients.

Ingredient Price per kilogram (RM)

Year 2004 Year 2005

P 5.00 w

Q 2.50 4.00

R x y

S 4.00 4.40

a) The index number of gradient P in the year 2005 based on the year 2004 is 120. Calculate the

value of w. [2 marks]P2,2006

b) The index number of gradient R in the year 2005 based on the year 2004 is 125. The price per

kilogram of ingredient R in the year 2005 is RM2.00 more than its corresponding price in the year

2004. Calculate the value of x and of y. [3 marks]P2,2006

c) The composite index for the cost of making the cake in the year 2005 based on the year 2004 is

127.5. Calculate

i) the price of a cake in the year 2004 if its corresponding price in the year 2005 is RM30.60

ii) the value of m if the quantities of gradients P, Q, R, and S used are in the ratio of 7 : 3 : m : 2.

[5 marks]P2,2006

P a g e | 28


4 Table shows the prices and the price indices of five components P, Q, R, S, and T, used to produce a

kind of toy.

Component Price (RM) for the year Price index for the year

2006 based on the year 2004 2004 2006

P 1.20 1.50 125

Q x 2.20 110

R 4.00 6.00 150

S 3.00 2.70 y

T 2.00 2.80 140

a) Find the value of x and of y. [3 marks]P2,2007

b) Calculate the composite index for the production cost of the toys in the year 2006 based on the year

2004. [3 marks]P2,2007

c) The price of each component increases by 20% from the year 2006 to the year 2008. Given that the

production cost of one toy in the year 2004 is RM55, calculate the corresponding cost in the year

2008.

[4 marks]P2,2007

S R

Q

P

T 72o 90

o

36o

144o

P a g e | 29


5 Table shows the prices and the price indices of four ingredients, fish, flour, salt and sugar used to make

a type of fish cracker.

Diagram shows a pie chart which represents the relative quantity of the ingredients used.

Ingredients Price (RM) per kg for the year Price index for the year 2005

based on the year 2004 2004 2005

Fish 3.00 4.50 150

Flour 1.50 1.80 h

Salt k 0.90 112.5

Sugar 1.40 1.47 105

a) Find the value of h and of k [3 marks]P2,2008

b) Calculate the composite index for the cost of making these crackers in the year 2005 based on the

year 2004. [3 marks]P2,2008

c) The composite index for the cost of making these crackers increases by 50% from the year 2005 to

the year 2009.

Calculate

i) the composite index for the cost of making these crackers in the year 2009 based on the year

2004.

ii) the price of a box of these crackers in the year 2009 if its corresponding price in the year 2004

is RM25.

[4 marks]P2,2008

Salt

15%

Sugar

10%

Fish

30%

Flour

45%

P a g e | 30


6 Table shows the prices, the price indices and weightages for four types of stationery P, Q, R and S.

Stationery Price (RM) per unit Price index for the year 2008

based on the year 2007 Weightage

Year 2007 Year 2008

P 2.80 2.10 x 4

Q 4.00 4.80 120 2

R 2.00 y 130 3

S z 5.80 116 m

a) Find the value of

i) x

ii) y

iii) z [3 marks]P2,2009

b) The composite index for the price of the stationery in the year 2008 based on the year 2007 is

108.4

Calculate the value of m [3 marks]P2,2009

c) The total expenditure for the stationery in the year 2007 is RM525.

Calculate the corresponding total expenditure in the year 2008. [2 marks]P2,2009

d) The price index for Q in the year 2009 based on the year 2007 is 132.

Calculate the price index for Q in the year 2009 based on the year 2008 [2 marks]P2,2009

P a g e | 31


7 Table shows the price indices for three items P, Q and R used in the production of a type of bag.

Item Price index in the year 2006


Price index in the year 2008


P 125 150

Q 116 x

R y 120

(a) Find the price index of item P in the year 2008 based on the year 2006.

[2 marks]P2,2010

(b) The price of item Q in the year 2004 is RM7.50 and its price in the year 2008 is RM10.50

Find

(i) the value of x,

(ii) the price of item Q in the year 2006

[3 marks]P2,2010

(c) The composite index for the production cost of the bag in the year 2006 based on the year 2004

is 118.5. The cost of the items P, Q and R used are in the ratio of 2 : 1 : 3

Find the value of y.

[3 marks]P2,2010

(d) Given the price of the bag in the year 2006 is RM47.40, find the corresponding price of the bag

in the year 2004.

[2 marks]P2,2010

P a g e | 32


MOTION ALONG A STRAIGHT LINE

1 A particle moves along a straight line from a fixed point P. Its velocity V m s-1

, is given by

V = 2t(6 – t), where t is the time, in seconds, after leaving the point P.

(Assume motion to the right is positive)

Find

a) the maximum velocity of the particle,

[3 marks]P2,2004

b) the distance travelled during the third second,

[3 marks]P2,2004

c) the value of t when the particle passes the point P again,

[2 marks]P2,2004

d) the time between leaving P and when the particle reverses its direction of motion.

[2 marks]P2,2004

2 Diagram shows the position and directions of motion of two objects, P and Q, moving in a straight line

passing two fixed points, A and B respectively. Object P passes the fixed point A and object Q passes

the fixed point B simultaneously. The distance AB is 28 m.

The velocity of vp m s-1

, is given vp = 6 + 4t -2t2, where t is the time, in seconds after it passes A,

while Q travels with a constant velocity of -2 m s-1

. Object P stops instantaneously at the point C.

(Assume that the positive direction of motion is towards the right)

Find

a) the maximum velocity in m s-1

, of P,

[3 marks]P2,2005

b) the distance, in m, of C from A,

[4 marks]P2,2005

c) the distance, in m, between P and Q when P is at the point C.

[3 marks]P2,2005

3 A particle moves in a straight line and passes through a fixed point O. Its velocity, v m s-1

, is given by

v = t2 – 6t + 5, where t is the time, in seconds, after leaving O.

[Assume motion to the right is positive.]

a) Find

i) the initial velocity of the particle,

ii) the time interval during which the particle moves towards the left,

iii) the time interval during which the acceleration of the particle is positive.

[5 marks]P2,2006

b) Sketch the velocity-time graph of the motion of the particle for 0 t 5

[2 marks]P2,2006

c) Calculate the total distance travelled during the first 5 seconds after leaving O.

[3 marks]P2,2006

P Q

A B C

28 m

P a g e | 33


4 A particles moves along a straight line and passes through a fixed point O. Its velocity, v m s-1

, is

given by v = t2 – 6t + 8, where t is the time, in seconds, after passing through O.

[Assume motion to the right is positive.]

Find

a) the initial velocity, in m s-1

[1 mark]P2,2007

b) the minimum velocity, in m s-1

[3 marks]P2,2007

c) the range of values of t during which the particle moves to the left.

[2 marks]P2,2007

d) The total distance, in m, travelled by the particle in the first 4 seconds.

[4 marks]P2,2007

5 A particle moves along a straight line and passes through a fixed point O. Its velocity, v m s-1

, is given

by v = 10 + 3t – t2, where t is the time, in seconds, after passing through O. The particle stops

instantaneously at a point R.

[Assume motion to the right is positive]

Find

a) the acceleration, in m s-2

, of the particle at R,

[3 marks]P2,2008

b) the maximum velocity, in m s-1

, of the particle,

[3 marks]P2,2008

c) the total distances, in m, travelled by the particle in the first 9 seconds, after passing through O.

[4 marks]P2,2008

6 A particle moves in a straight line and passes through a fixed point O. The velocity of the particle,

v cm s-1

, is given by v = 15 + 4t – 3t2, where t is the time in s, after leaving O

Find

a) the initial velocity, in cm s-1

, of the particle,

[1 mark]P2,2009

b) the initial acceleration, in cm s-2

, of the particle,

[2 marks]P2,2009

c) the maximum velocity, in cm s-1

of the particle,

[3 marks]P2,2009

d) the distance, in cm, from O when the particle stops instantaneously.

[4 marks]P2,2009

7 A particle moves along a straight line such that its acceleration, a ms-2

, is given by a = 10 – 2t, where t

is the time, in seconds, after it passes through a fixed point O. The initial velocity of the particle is -

16ms-1

.

[Assume motion to the right is positive]

Find

(a) the maximum velocity, in ms-1

, of the particle, [4 marks]P2,2010

(b) the range of values of t during which the particle moves to the right, [2 marks]P2,2010

(c) the total distance, in m, travelled by the particle in the first 6 seconds. [4 marks]P2,2010

P a g e | 34


LINEAR PROGRAMMING


A district education office intends to organise a course on the teaching of Mathematics and Science in

English.

The course will be attended by x Mathematics participants and y Science participants.

The selection of participants is based on the following constraints.

I : The total number of participants is at least 40.

II : The number of Science participants is at most twice that of Mathematics.

III : The maximum allocation for the course is RM7 200. The expenditure for a Mathematics

participant is RM120 and for a science participant is RM80.

(a) Write down three inequalities, other then x 0 and y 0, which satisfy the above constraints.

[3 marks]P2,2004

(b) Hence, by using a scale of 2 cm to 10 participants on both axes, construct and shade the region R

which satisfies all the above constraints.

[3 marks]P2,2004

(c) Using your graph from (b), find

(i) the maximum and minimum number of Mathematics participants when the number of

Science participants is 10.

(ii) the minimum cost to run the course.

[4 marks]P2,2004


An institution offers two computers courses, P and Q. The number of participants for course P is x and

for course Q is y.

The enrolment of the participants is based on the following constraints.

I : The total number of participants is not more than 100.

II : The number of participants for course Q is not more than 4 times the number of participants

for course P by at least 5.

III : The number of participants for course Q must exceed the number of participants for course

P by at least 5.


[3 marks]P2,2005

(b) By using a scale of 2 cm to 10 participants on both axes, construct and shade the region R that

satisfies all the above constraints.

[3 marks]P2,2005

(c) Using your graph from (b), find

(i) the range of the number of participants for course Q if the number of participants for

course P is 30.

P a g e | 35


(ii) the maximum total fees per month that can be collected if the fees per month for

courses P and Q are RM50 and RM60 respectively.

[4 marks]P2,2005


A workshop produces two types of rack, P and Q. The production of each type of rack of rack involves

two processes, making and painting. Table shows the time taken to make and paint a rack of type P

and a rack of type Q.

Rack Time taken (minutes)

Making Painting

P 60 30

Q 20 40

The workshop produces x racks of type P and y racks of type Q per day. The production of racks per

day is based on the following constraints.

I : The maximum total time for making both racks is 720 minutes.

II : The total time for painting both racks is at least 360 minutes.

III : The ratio of the number of racks of type P to the number of racks of type Q is at least 1:3


[3 marks]P2,2006

(b) Using a scale of 2 cm to 2 racks on both axes, construct and shaded the region R which satisfies all

of the above constraints.

[3 marks]P2,2006

(c) By using your graph in (b), find

(i) the minimum number of racks of type Q if 7 racks of type P are produced per day,

(ii) the maximum total profit per day if the profit from one rack of type P is RM24 and

from one rack of type Q is RM32.

[4 marks]P2,2006


A factory produce two components, P and Q. In a particular day, the factory produces x pieces of

components P and y pieces of components Q. The profit from the sales of a piece of component P is

RM15 and a piece of component Q is RM12.

The production of the components per day is based on the following constraints.

I : the total numbers of components produced is at most 500

II : The numbers of component P produced is not more than three times the number of

component Q.

III : The minimum total profit for both components is RM4 200.


[3 marks]P2,2007

(b) By using a scale of 2 cm to 50 components on both axes, construct and shade the region R that

satisfies all the above constraints.

[3 marks]P2,2007

(c) Use your graph in (b), to find

P a g e | 36


(i) the minimum number of pieces of components Q if the number of pieces of component

P produced on a particular day is 100.

(ii) the maximum total profit per day.

[4 marks]P2,2007


The members of a youth association plan to organise a picnic. They agree to rent x buses and y vans.

The rental of a bus is RM800 and the rental of a van is RM300.

The rental of the vehicles for the picnic is based on the following constraints.

I : The total number of vehicles to be rented is not more than 8.

II : The number of buses is at most twice the numbers of vans.

III : The maximum allocation for the rental of the vehicles is RM4000.


[3 marks]P2,2008

(b) Using a scale of 2 cm to 1 vehicle on both axes, construct and shade the region R that satisfies all

the above constraints.

[3 marks]P2,2008

(c) Use the graph in (b) to find

(i) the minimum number of vans rented if 3 buses are rented.

(ii) the maximum number of members that can be accommodated into the rented vehicles if

a bus can accommodate 48 passengers and a van accommodate 12 passengers.

[4 marks]P2,2008


Johan uses x small tiles and y big tiles for decoration of his bathroom. The decoration budget is based

on the following constraints.

I : The total number of tiles cannot exceed 100 pieces.

II : The number of small tiles cannot exceed three times the number of big tiles.

III : The number of big tiles cannot exceed the number of small tiles by more than 30

pieces.


[3 marks]P2,2009

(b) Using a scale of 2 cm to 10 tiles on both axes, construct and shade the region R that satisfies all the

above constraints.

[3 marks]P2,2009

(c) Using the graph in (b), find

(i) the maximum number of small tiles that can be used.

(ii) the maximum total cost for the tiles if the cost of a small tile is RM1.50 and the cost of

a big tile is RM3.00

[4 marks]P2,2009

P a g e | 37



A training centre offers two courses, A and B. The number of participants for course A is x and the

number of participants for course B is y. The intake of the participants is based on the following

constraints :

I The maximum number of participants is 80.

II The number of participants for course B is at least 10.

III The number of participants for course B is at most 2

3 times the number of

participants for course A.


[3 marks]P2,2010

(b) Using a scale of 2 cm to 10 participants on both axes, construct and shade the region R which

satisfies all of the above constrains.

[3 marks]P2,2010

(c) Using the graph constructed in (b), find

(i) the minimum number of participants for course A.

(ii) the maximum total fees collected per month if the monthly fees per participant for course A is

RM300 and for course B is RM400.

[4 marks]P2,2010

P a g e | 38


SECTION A

FUNCTIONS

1a) 3

2x

1b) 15

15x

1c) 410)( xxh

QUADRATIC FUNCTIONS

1a) (0 , -5)

1b) k = 4 , p = -1

2ai) h = 3 , k = 2

2aii) x < 2 , x > 3

2b) x2 -9x + 20 = 0

3a) 3x + 2y + 23 = 0

3bi) B(0 , -5)

3bii) C(2

21,2)

SIMULTANEOUS EQUATIONS

1) m = 0.606 , -6.606 , p = 2.606 , -4.606

2) y = -4 , 3 , x = 3 , 2

1

3) x = -0.693 , 1.433 , y = 2.386 , -1.886

4) x = 3 , 1 , y = 3 , -1

5) x = 5 , -6 , y = 3, 3

2

6) p = 1.577 , 0.423 , k = 3.731 , 0.269

7) x = 13.24 , 4.76 , y = 3.12 , -1.12

COORDINATE GEOMETRY

1a) 169

yx

1b) D(3 , -4)

1c) 2

1

2a) (-3 , -2)

2b) 2y = x – 1

STATISTICS

1a) mean = 15 , variance = 22.2

1bi) 26

1bii) 5.494

2a) 24.07

2b) 11.74

3a) k = 4

3b) 43

3c) 48

4a)

Score 0-9 10-19 20-29 30-39 40-49

No. of

students 4 6 10 8 4

4b) 18.33

5) x = 13 , y = 5

6a) 26

6b) 11.43

PROGRESSIONS

1a) 4

1r

1bi) Triangle number 5 , T5

1bii) 21333

1

2a) 294

2b) RM1019.20

3a) k = 100 , m = 20

3b) n = 11

4ai) 415 cm

4aii) 14 000

4b) n = 33

5a) RM22 973

5b) n = 16

5c) RM122 434

6a) 7

6b) 1 800 cm

7a) 324

7b) n = 12

VECTORS

1ai) -2y + 6x

1aii) xy2

9

2

3

P a g e | 39


1bi) h(6x – 2y)

1bii) )2

9

2

3( xyk

1c) h = 2

1 , k =

3

1

2ai) -20x + 32y

2aii) 25x

2b) BD = 4BF

2c) 104

3a) 5x + 2y

3bi) 4x + my

3bii) m = 5

8

4ai) x – 3y

4aii) 3

2x + y

4b) h = 2

1 , k =

3

4

5ai) -3x + 4y

5aii) x + 3

8y

5b) h = 11

9 , k =

11

3

TRIGONOMETRIC FUNCTIONS

1a,b)

2sin2x = 2 -

180

x

1 – 2sin2x = 1 – (2 -

180

x)

cos2x = 180

x - 1

y = 180

x - 1

Number of solutions = 2

2a) cosec2x – 2sin

2x – cot

2x

= 1 + cot2x – 2sin

2x – cot

2x

= 1 – 2sin2x

= cos2x

2b)

3(cosec2x – 2sin

2x – cot

2x) = 1

x

3cos2x = 1

x

cos2x = 3

1

3

x

y = 3

1

3

x


3a)

3b) x

+ 2cos x = 0

-2cos x = x

1180

x

y

y= 3

1

3

x

y = x

P a g e | 40


y = x


4a,b)

2 - |3cos2x| = 2

x

|3cos2x| = 2 - 2

x

y = 2 - 2

x, Number of solutions = 8

5a,b) x

x2sec2

tan2

=

)tan1(2

tan22 x

x

= x

x2tan1

tan2

= tan 2x

x3+ 0

sec2

tan22

x

x

x3 + tan2x = 0

-tan2x =

x3

y =

x3, Number of solutions = 3

6a,b)

x3

4-cos2x =

2

3

cos2x = x3

4-

2

3

2

3cos2x =

2

3( x3

4-

2

3)

y = 4

92x


7a,b)

6cos x = 4 - 3x

3cosx =

2

4 - x

2

3

1 + 3cosx = 2 - x2

3 + 1

1 + 3cosx = 3 - x2

3

y

x

4

2

4

3

3

xy

3

y = 2 - 2

x

y = 4

92x

y

x

-2

3/2 2 /2 0

3

4

y = 3 - x2

3

y = |3 cos2x|

P a g e | 41


y = 3 - x2

3



1a) y = x3 – 3x

2 – 10

1b) (2 , -14) minimum point

2a) h = 2 , k = 6

2b) y = 533

2 23 xx

3ai) - 10

3aii) 24

3b) g(x) = x2 – 8x + 7

SECTION B

COORDINATE GEOMETRY

1ai) y = 2x + 17

1aii) B (-8 , 1)

1b) D (-14 , -11)

1c) x2 + y

2 + 8x – 18y + 72 = 0

2a) 9 unit2

2b) C (5

2,

5

12)

2ci) x2 + y

2 – 18x + 8 y + 45 = 0

2cii) Shown : b2 – 4ac < 0

3a) 4x2 + 4y

2 – 24x – 8y +15 = 0

3bi) k = 3

3bii) Q (2

9, -1)

3c) 7.5 unit2

4ai) k = 4

4aii) A(-2 , 3)

4bi) B(4 , 7)

4bii) 2y + 3x = 26

4c) 3x2 + 3y

2 + 24x -10y – 13 = 0

CIRCULAR MEASURE

1a) 3

2 radian

1b) 20.95 cm

1c) 61.40 cm2

2a) 6 cm

2b) 24.40 cm

2c) 37.46 cm2

3a) 62.59 m2

3b) 38.25 m

3c) 31.39 m2

4a) 31.42 cm

4b) 88.63 cm2

5a) cos = 5

1

= 1.37 radian, shown

5b) 14.18 cm

5c) 40.66 cm2

6a) 0.39 radian

6b) 40.00 cm

6c) 82.18 cm2

7a) 4.189 radian

7b) 10 cm

7c) 61.89 cm

7d) 61.43 cm2

DIFFERENTIATION

1a) 3dx

dy

1b) x – 3y + 7 = 0

1c) Q(1 , 5) maximum point


1a) y = -12x + 15

1bi) 5

1

1bii) 1125

49

2a) k = 5

2b) 206

5

2c) 65

2

3a) k = 7

3b) (1 , 8) minimum point

3c) y = x2 +

x

2+ 5

4a) k = 2

4bi) 8

4bii) 7

4

P a g e | 42


5a) y = 2x + 4

5b) 3

1

5c) 2

LINEAR LAW

1a,b)

x 2 4 6 8 10 12

log10y 0.50 0.74 0.96 1.21 1.46 1.67

y = pk

x

log10y = log10p + xlog10k

log10y = (log10k)x + log10p

log10p = 0.26

p = 1.820

log10k = Gradient of the graph

= 0.1161

k = 1.306

2a,b)

x2 1 4 9 16 25 30.25

xy 5.5 9.4 15.0 26.0 38.5 46.2

y = px + px

r

xy = px2 +

p

r

p = gradient of the graph

= 1.375

p

r= Y-intercept

375.1

r= 4

r = 5.5

3a,b)

x + 1 2 3 4 5 6 7

log y 0.60 0.76 0.94 1.12 1.30 2.46

y = pkx+1

log y = log p + (x + 1)log k

log y = (log k)(x + 1) + log p

log p = Y- intercept

log p = 0.26

p = 1.82

log k = Gradient of the graph

= 0.172

k = 1.486

P a g e | 43


4a,b)

x 2 3 4 5 6 7

x

y

4 4.4 5 5.5 6.1 6.5

y = 2kx2 + x

k

p

x

y = 2kx +

k

p

2k = Gradient of the graph

= 0.5

k = 0.25

k

p = Y-intercept

25.0

p = 3

p = 0.75

5a,b)

x 1.5 3.0 4.5 6.0 7.5 9.0

log y 0.40 0.51 0.64 0.76 0.89 1.00

ci) When y = 4.8, log10y = 0.68

Based on the graph, x = 5

y = hk2x

log10y = log10h + 2xlog10k

log10y = (2log10k)(x) + log10h

log10h = Y-intercept

= 0.28

h = 1.91

2log10k = Gradient of the graph

= 0.08

log10k = 0.04

k = 1.10

6a,b)

x

1 0.67 0.50 0.33 0.25 0.20 0.17

y

1 0.40 1.30 2.15 2.60 2.85 3.05

P a g e | 44


y

k=

x

p+ 1

y

1= (

k

p)(

x

1) +

k

1

k

1 = Y-intercept

= 3.9

k = 0.26

k

p = Gradient of the graph

26.0

p = -5.213

p = -1.34

7a,b)

x 3 4 5 6 7 8

log y 0.41 0.52 0.61 0.69 0.80 0.90

y = k

h x

log y = xlog h – log k

log y = (log h)(x) – log k

log h = Gradient of the graph

= 0.098

h = 1.253

- log k = Y-intercept

= 0.11

log k = -0.11

k = 0.7762

VECTOR

1ai) 2x – 6y

1aii) 4x + 3y

1b) h = 5

2 , k =

5

4

1c) 24.08

2ai) 3

2x – y

2aii) 2

1x +

2

1y

2bi) 2

1hx +

2

1hy

2bii) 3

2kx + (1 – k)y

2c) h = 5

4 , k =

5

3

PROBABILITY DISTRIBUTIONS

1ai) p = 0.6

1aii) 0.9993

1bi) 0.2743

1bii) m = 67.63

2ai) 0.294

2aii) 0.797

2bi) 0.945

2bii) m = 3.07

3a) 0.2119

3b) 246

3c) 4.2

4ai) 0.2090

4aii) 0.6846

4bi) 0.7357

4bii) 1250

5a) 0.9505

5bi) 475

5bii) m = 209.84

P a g e | 45


6ai) 10

6aii) 2.739

6bi) 0.01622

6bii) 0.7560

7ai) 5

2

7aii) 0.1673

7bi) 0.3203

7bii) 2.54

SECTION C

SOLUTION OF TRIANGLES

1ai) 57.23o

1aii) 106.07o

1aiii) 82.37 cm2

1bi)

1bii) 122.8o

2a) 19.27 cm

2b) 50o44 , 129

o16

2ci) 24.89 cm

2cii) 290.1 cm2

3a) 60o4

3b) 5.573 cm

3c) 116o33

3d) 35.43 cm2

4ai) 13.36 cm

4aii) 23o53

4bi)

4bii) 13.80 cm2

5ai) 19.34

5aii) 16.21

5b) 10.50

5c)

6a) 23.23

6b) 21.76

6c) 98o11

6d) 60.31 cm2

7ai) 2.207

7aii) 7.072

7bi)

7bii) 75o

7biii) 8.264

INDEX NUMBER

1ai) RM29.00

1aii) 162

1bi) 125

1bii) RM25.00

2a) x = 125 , y = 2.80 , z = 0.50

2bi) 129.44

2bii) RM2 306.09

2c) 194.16

3a) 6.00

3b) x = 8 , y = 10

3ci) RM24.00

3cii) m = 4

4a) x = 2.00 , y = 90

4b) 123.5

4c) RM81.51

5a) h = 120 , k = 0.80

5b) 126.38

A B

C

B

B A

C

23o53

A

B C 45o

35 30.21

C C B

A

4 cm

50o

P a g e | 46


5ci) 189.57

5cii) RM47.39

6ai) x = 75

6aii) y = 2.60

6aiii) z = 5.00

6b) m = 6

6c) RM569.10

6d) 110

7a) 120

7bi) 140

7bii) RM8.70

7c) 115

7d) RM40.00

MOTION ALONG A STRAIGHT LINE

1a) 18 ms-1

1b) 17 3

1m

1c) t = 9 s

1d) 6 s

2a) 8 ms-1

2b) 18 m

2c) 4 m

3ai) 5 ms-1

3aii) 1 < t < 5

3aiii) t > 3 s

3b)

3c) 13 m

4a) 8 ms-1

4b) -1 ms-1

4c) 2 < t < 4

4d) 8 m

5a) -7 ms-2

5b) 12.25 ms-1

5c) 1236

1m

6a) 15 cm s-1

6b) 4 cm s-2

6c) 163

1 cm s

-1

6d) 36 cm

7a) 9 ms-1

7b) 2 < t < 8

7c) 413

1 m

LINEAR PROGRAMMING

1a)

I : x + y ≥ 40

II : y ≤ 2x

III : 3x + 2y ≤ 180

ci) When y = 10

x min = 30

x max = 53

cii) When x = 14, y = 26

minimum cost = 120(14) + 80(26)

= RM3760

5

-4

0 1 5 3

V/ms-1

t/s

y = 2x

(14,27)

R 3x + 2y = 180

x + y = 40

P a g e | 47


2a,b)

I : x + y ≤ 100

II : y ≤ 4x

III : y ≥ x + 5

ci) The range of the number of participants for

course Q is 35 ≤ y ≤ 70

cii) Objective function = 50x + 60y

Let the total fees collected be RM3000

50x + 60y = 3000

5x + 6y = 300

Based on the graph, the maximum

collection is obtained when

x = 20 and y = 80

Maximum total fees

= 50(20) + 60(80)

= RM5800

3a,b)

I : 60x + 20y ≤ 720

II : 3x + 4y ≥ 36

III : y ≤ 3x

ci) Based on the graph, when x = 7, the minimum

value of y = 4

The minimum number of racks of tape Q produce

per day is 4.


Let the profit be RM192

24x + 32y = 192

3x + 4y = 24

Based on the graph, the maximum profit is

obtained when x = 6 and y = 18.

Maximum profit = 24(6) + 32(18)

= RM720

4a,b)

I : x + y ≤ 500

II : y ≥ 3

1x

III : 5x + 4y ≥ 1400

R

y = 4x

x + y = 100

y = x + 5

5x + 6y = 300

(20,80)

3x+4y =24

3x+4y =36

3x+ y =36

y = 3x

(6,18)

R

P a g e | 48


ci) Based on the graph, when x = 100, the

minimum value of y = 225

The minimum number of pieces of component Q

produced is 225.


Based on the graph, the maximum total profit is


Maximum total profit

= 15(375) + 12(125)

= RM7 125

5a,b)

I : x + y ≤ 8

II : y ≥ 2

1x

III : 8x + 3y ≤ 40

ci)Based on the graph, when x = 3, the minimum

value of y = 2.

The minimum number of vans rented is 2.

cii) Let the number of members that can be

accommodated be 48.

48x + 12y = 48

4x + y = 4

Based on the graph, the maximum number of

members that can be accommodated is occurred

when x = 4 and y = 2.

Maximum number of members

= 48(4) + 12(2)

= 216

6a,b)

I : x + y ≤ 100

II : y ≥ 3

1x

III : y ≤ x + 30

y = 3

1x

5x + 4y = 1400

x + y = 500

(375,125)

R R

(4,2)

y = 2

1x

x + y = 8

8x + 3y = 40

4x + y = 4

P a g e | 49


ci) Based on the graph, the maximum number of

small tiles that can be used is 75.

cii) Objective function = 1.5x + 3y

Let the total cost of the tiles be RM60.

1.5x + 3y = 60

x + 2y = 40

Based on the graph, the maximum total cost is


Maximum total cost

= 1.5(35) + 3(65)

= RM247.50

7a,b)

I: x + y ≤ 80

II : y ≥ 10

III : y ≤ 2

3x

ci) Based on the graph, the minimum number of

participants for course A is 7.


Let the total fees collected per month be RM12 000

300x + 400y = 12 000

3x + 4y = 120

Based on the graph, the maximum total fees are

obtained when x = 32 and y = 48

Maximum total fees collected per month

= 300(32) + 400(48)

= RM28 800

R x + 2y = 40

y = x + 30

x + y = 100

y = 3

1x

(35,65) x + y = 80

y = 2

3x

y = 10

R

(32,48)

3x + 4y = 120

Kertas 2 SPM Matematik Tambahan 2004-2010

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Transcript of Kertas 2 SPM Matematik Tambahan 2004-2010